ab initio study of the structural, electronic, and...

Turk J Elec Eng & Comp Sci

(2018) 26: 1249 – 1260

c⃝ TUBITAK

doi:10.3906/elk-1711-106

Turkish Journal of Electrical Engineering & Computer Sciences

http :// journa l s . tub i tak .gov . t r/e lektr ik/

Research Article

Ab initio study of the structural, electronic, and magnetic properties of Co2FeGa

and Co2FeSi and their future contribution to the building of quantum devices

Abdallah OUMSALEM1,∗, Yamina BOUREZIG2, Zakia NABI3, Badra BOUABDALLAH3

1Laboratory of Theoretical Physics and Material Physics, Hassiba Ben Bouali University of Chlef, Chlef, Algeria2Elaboration and Characterization Laboratory, Department of Electronics, University Djillali Liabes,

Sidi Bel Abbes, Algeria3Condensed Matter and Sustainable Development Laboratory, Department of Physics, University of Sidi-Bel-Abbes,

Sidi-Bel-Abbes, Algeria

Received: 12.11.2017 • Accepted/Published Online: 03.04.2018 • Final Version: 30.05.2018

Abstract: How small can an electronic device be and still function? How many atoms are needed for such device?

At what point will the semiconductor fabrication technology be unable to construct anything smaller? These are

common questions asked by researchers, and the answers indicate the short-term limitations on the use of semiconductor

technology. When we reach these limitations, we can propose two methods to solve these problems: first, we could

continue using semiconductor technology, alongside the development of new theories and techniques to counter quantum

effects caused by the miniaturization of electronic devices like transistors and microprocessors. Second, we could exploit

these quantum effects to invent a new generation of electronic devices. Doing that, we propose spintronics or the use

of spin properties. Heusler compounds as cobalt base alloys (Co2YZ) present particular interest for spin electronics

applications. In this paper, we present properties and results for two cobalt base alloys, Co2FeGa and Co2FeSi. These

properties are interesting for the field of quantum computation. In the first part of this paper we introduce Moore’s law,

which explains the major limits of semiconductor technology. Then we discuss the results of our calculations based on

the use of density functional theory and the WIEN2K program. This is for the purpose of making new quantum devices.

Key words: Heusler alloys, Co2FeGa, Co2FeSi, spin polarization, quantum devices

1. Introduction

To date, semiconductor technology has been the preferred method for the construction of circuits, chips, and

computers. However, it is unclear how much longer we can continue to develop semiconductors before quantum

effects become problematic. This was predicted by semiconductor industry experts in their National Technology

Roadmap for Semiconductors [1], Gordon Moore [2], and other researchers [3–10]. It has been suggested that a

solution may be found in the use of quantum computation techniques based on other kinds of materials such as

Heusler alloys. The structural, electronic, and magnetic characteristics of ferromagnetic alloys like high-speed

polarization and magnetic moment help to construct a new generation of quantum devices. The field of quantum

computers had its first steps in the early 1980s, in the work of Paul Benioff [11] and Richard Feynman [12,13].

Subsequent attempts at the design of quantum algorithms included work by Deutsch and Jozsa [14], Simon

[15], and Shor [16,17]. After that, the first 2-qubit quantum computer was built [18]. There are a number

of additional textbook examples that demonstrate the power of quantum computation over semiconductor

∗Correspondence: [email protected]

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OUMSALEM et al./Turk J Elec Eng & Comp Sci

technology using classical computation methods [19–29]. Among the various half metallic materials proposed

to date, such as double perovskite, spinel, or zinc-blende structured materials, Heusler alloys have recently

attracted much attention for their relatively high spin polarization and large magnetic moment. On the base of

an ab initio study on the structural, electronic, and magnetic properties of the two alloys Co2FeGa and Co2FeSi,

we want to improve how quantum computation will be possible in the near future. Besides the construction of

a quantum bit (Qubit), the use of the high spin polarization of the Heusler compounds offers an opportunity

for the construction of a set of universal quantum gates such as QNOT, Hadamard, Y-Pauli, and Z-Pauli gates

[30–36]. These are based on the use of the quantum superposition of electron orbits and spin polarization

[37]. As a result of using developed techniques to obtain the electron superposition, the fabrication and the

characterization of quantum dots and gates are possible [38]. The characteristic of quantum superposition

of states in the quantum gates offers extensions in memory and speed. However, this is not the case in the

classical gates (Not, AND, OR, NAND, XOR) and circuits based on semiconductor materials [39]. This needs

more transistors in a chip to extend memory and accelerate computation. Based on density functional theory

(DFT) and the WIEN2K program, we study if the properties of Heusler compounds respond to the needs of

quantum device construction. Besides the use of generalized gradient approximation (GGA) to determine the

partial and total magnetic moment, we study the spin polarization of the two materials, Co2FeGa and Co2FeSi.

2. Moore’s law and the limits of semiconductor technology

How will quantum computing techniques end the power of the use of semiconductor technology? This is a

big question for the contemporary electronic devices industry. After the invention of the transistor in 1947,

technology rapidly progressed over the next six decades. This has led to progressively more sophisticated

generations of microprocessors and computers. However, according to Moore’s law and other researchers, it is

possible that this progress may one day end.

From his experience in the employment of Intel, Gordon Moore forecasted the rapid development of

semiconductor technology. Moore’s law stated that the transistor density on integrated circuits doubles about

every 2 years. As a result, we have obtained powerful computers with high functionality and performance, as

shown in Table 1.

In Table 1 we see how the number of transistors increases with each new generation of microprocessors.

The first microprocessor 4004 included 2300 transistors on a relatively large chip. However, in the latest

generations of microprocessors such as Intel Itanium and Intel Itanium 2 there are more than 100 million

transistors contained on a very small chip. Thus, we can extrapolate to a future microprocessor generation

with more than 100 billion transistors on a micrometer chip. Furthermore, Moore’s predictions point towards

a future limit on semiconductor technology. According to Moore, semiconductor technology will approach the

atomic scale. Thus, we have many difficulties in the development of exponential trends such as functionality and

performance. Thinking about other material is a necessity. Ferromagnetic materials have different characteristics

than semiconductor materials. Heusler alloys, a kind of ferromagnetic material, have high spin polarization and

magnetic moment, which will be proven in the next parts of this work. This allows electrons to occupy more

states than their original states. The superposition of electrons states can extend the performance and the speed

of electronic devices. For the two Heusler compounds Co2FeGa and Co2FeSi spin (up and down) polarization

represents the superposition of electrons. This is the representation of the quantum bit, which is the smallest

unit of quantum information. With the use of the quantum computation, we can assure a doubling of speed

and performance every 18 months or less. Thus, Moore’s law returns to play a great role in the determination

of the future generation of computers and microprocessors.

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Table 1. The integration rate of transistors in microprocessor generations [1].

Microprocessor Year of introduction Transistors

4004 1971 2300

8008 1872 2500

8080 1974 4500

8086 1978 29,000

Intel286 1982 134,000

Intel386TM processor 1985 275,000

Intel486TM processor 1989 1,200,000

Intel Pentium Processor 1993 3,100,000

Intel Pentium II Processor 1997 7,500,000

Intel pentium III Processor 1999 9,500,000

Intel Pentium 4 Processor 2000 42,000,000

Intel Itanium Processor 2001 25,000,000

Intel Itanium 2 Processor 2002 220,000,000

Intel Itanium 2 Processor (9 Mb cache) 2004 592,000,000

3. The DFT method and the WIEN2K program

3.1. The DFT theory

DFT is one of the most widely used methods in ab initio calculations of the structure of atoms, molecules,

crystals, and surfaces. Furthermore, it is a standard method of calculation used for solving many-body quantum

problems of the types encountered in the studies of correlated polyelectronic systems in general and crystalline

solids in particular. A first approach was proposed by Thomas and Fermi in the 1920s. Then the theory was

developed on the basis of the Hohenberg–Kohn theorems, which are relative to any electron (fermion) system

in an external field such as that induced by the nuclei and going beyond the Hartree–Fock approximation

through taking into account correlation effects in the studies of the ground state physical properties of correlated

polyelectronic systems. In addition, the corrections introduced in terms of exchange-correlation contributions

have revealed better precision of the polyelectronic systems’ energies calculations [40].

The bodies in the crystal structure are atoms (nuclei, electrons). The Hamiltonian of the system composed

of electrons and nuclei is written in the following form:

H = Te + Tn + Ue−e + Ue−n + Un−n, (1)

where Te and Tn refer to the kinetic energy operators of electrons and nuclei, and the three other terms

Ue−e, Ue−nUn−n represent the interaction between electrons and electrons, electrons and nuclei, and nuclei and

nuclei, respectively.

3.2. The WIEN2K program

The simulation code WIEN was developed at the Institute of Materials Chemistry at the Technical University

of Vienna and was published by Peter Blaha and Karlheinz Schwarz. This code has been continuously revised

and has experienced several updates. Versions of the WIEN code have been developed, named according to the

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year of their publication. such as WIEN93, WIEN95, and WIEN97. In our study we use the version WIEN2K.

WIEN2K has powerful characteristics, particularly in terms of speed and universality (multiplatform). The

WIEN2K package is written in FORTRAN90 and runs on a UNIX operating system. Moreover, it consists

of several independent programs, which perform electronic structure calculations of solid bodies based on

DFT. Several properties of materials can be calculated with this code such as energy bands, density of states,

Fermi surface, electron density, spin density, X-ray structure factors, total energy, atomic forces, equilibrium

geometries, optimizations of structure, electric field gradients, isometric offsets, hyperfine fields, polarization of

spins (ferromagnetic, antiferromagnetic, or other structures), spin-orbit coupling, X-ray emission and adsorption

spectra, and optical properties. In our paper we depend on WIEN2K to achieve the different characteristics of

the two ferromagnetic Heusler alloys Co2FeGa and Co2FeSi.

3.3. Heusler alloys

The full Heusler alloys crystallize in the structure L21 , and they have a stoichiometric composition of type

X2YZ, where X and Y are transition metals and Z represents the nonmagnetic element of the groups III, IV, or

V in the periodic table. In general, the Heusler alloys crystallize in cubic structures of the face-centered cubic

(fcc) Bravais lattice, which show centered faces (space group Fm-3, No. 255), where the X atoms occupy sites

A (1/4, 1/4, 1/4) and C (3/4, 3/4, 3/4), the Y atom occupies site B (0, 0, 0), and the Z atom occupies site D

(1/2, 1/2, 1/2). Figure 1 shows the full Heusler alloys of type X2YZ structural characterization.

Figure 1. Schematic representation of the L21 structure for full Heusler alloys of type X2YZ.

Cobalt-based alloys (Co2YZ) [41] are theoretically predicted to have a semimetallic character at room

temperature. Therefore, they present particular interest for tspin electronics applications. In addition, these

materials have a Curie temperature much higher than room temperature [42], with values up to 1120 K in

Co2FeSi [43]. Moreover, they present a good lattice mismatch (epitaxy) with that of the MgO substrate. This

good epitaxy between the layer and the substrate leads to a significant improvement in magnetic properties of

these systems [44].

4. Calculation details

The calculations have been carried out based on DFT [45] and with the use of the WIEN2K program [46,47],

which is an implementation of the FP-LAPW method [48,49]. The exchange and correlation potential is treated

with GGA as presented by Perdew et al. [50–53].

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The first step is to make a precise choice of important parameter values such as muffin-tin radius Rmt

and the Brillouin zone of the various atoms constituting the compounds studied. Rmt is an approximation

proposed by John C Slater and it is a form approximation of the potential field in an atomistic environment.

Moreover, the Brillouin zone was developed by Leon Brillouin, and it is the set of points enclosed by the Bragg

planes. We also call it the Wigner–Seitz cell.

We choose our Rmt values as follow: for the cobalt atom 2.2 UA, for the iron atom 2.0 UA, and for the

silicon and the gallium a value of 1.9 UA has been chosen.

The sampling of the Brillouin zone must be done with care using the special point’s technique of the

Monkhorst and Pack lattice [54]. In our case, we use 10 special points, which are able to ensure the convergence

of total energy by the mismatch of crystal.

These choices permit us to ensure the integration of the majority of core electrons in the sphere and to

avoid overlapping spheres (muffin-tin).

5. Results and discussion

We calculate the total magnetic moment, and in each atomic sphere we use GGA for the two compounds.

Moreover, we calculate the spin polarization of the two alloys Co2FeGa and Co2FeSi at an energy E (in

particular at the Fermi level EF , and in relation to the electronic spin-dependent densities of state (DOSs)).

The spin polarization that measures the spin asymmetry is given by the following expression [55]:

P (EF ) =n ↑ (EF )− n ↓ (EF )

n ↑ (EF ) + n ↓ (EF )(2)

where n ↑(EF ) and n ↓(EF ) are the majority and the minority densities of states at the Fermi level.

The values of total and partial magnetic moments of the two compounds as well as the polarization are

shown in Table 2. We notice that the major part of these magnetic moments is strongly localized in the site of

the element (Co) with a low contribution of Fe and the atoms of Si and Ga. The calculated magnetic moments

are in a good agreement with the available results.

Table 2. The calculated value of the total magnetic moment, the partial magnetic moment in µB , and the spin

polarization of the Heusler alloys Co2FeSi and Co2FeGa.

Compound mtotal (µB) mCo (µB) mFe (µB) mSi (µB) mGa (µB) P (%)

Co2FeSi 5.42035 1.36471 2.74021 –0.00054 - 63.22

6c

5.08d

Co2FeGa 5.00434 1.19597 2.76564 - –0.02673 96.89

5.08a

5.02b 1.22b 2.76b - –0.028b

a,b,cRef [56,57]. dRef [58].

We can see that for the studied alloys, the spin polarization at the Fermi level is greater than 95% and

can reach up to 100% for a slight variation of the lattice parameter, which is the case for the Co2FeGa alloy.

That said, the value of the spin polarization for the alloy can be explained by the gap absence in the majority

spin band at the Fermi level (in this study, the majority spin band is the spin-up part), as can be seen clearly

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in Figure 2a. Figure 2b represents the electronic DOS of the different energy level of the Co2FeSi alloy. It is

easy to see that the electronic structures for both alloys are approximately similar. They all have a forbidden

bandwidth in one spin direction. Also, it is clear from these figures that we have the presence of some peaks

around the Fermi level. Thus, we can assume that the DOS at the Fermi level is essentially linked to the Co

atom. Furthermore, the gap width is 0.45 eV for Co2FeGa and 0.49 eV for Co2FeSi. It has been found that in

Co2FeZ (Z = Si, Ga) the width of the band gap decreases with the increase of the atomic number of element

Z. Thus, the decrease in gap width can be explained in two ways. The first is the hybridization of p-d states.

The second is the change of the lattice parameter. The width of the gap is very sensitive to the change of the

lattice parameter and it decreases with the expansion of this parameter. In addition, we can note that, for the

two materials in our study, the magnetic moments of the Z elements are very small (less than 0.086 µB) at

0.086 for Ga and 0.050 for Si. They have the same sign as the element Fe, and they are antiparallel to the

magnetic moment of Co. Therefore, there is a double effect of the Z element in Co2FeZ. First, it determines

the lattice parameter of the alloy, and therefore the degree of localization of the electrons. Second, the total

magnetic moment is determined by the number of electrons supplied.

-20 -15 -10 -5 0 5

-3

-2

-1

0

1

2

3

Co2FeGa

DO

S (

Sta

tes/

eV)

Energy (eV)

up

dn

-20 -15 -10 -5 0 5

-3

-2

-1

0

1

2

3

Co 2FeGaD

OS

(S

tate

s/eV

)

Energy (eV)

up

dn

(b)(a)

Figure 2. Electronic density of state for the alloys Co2FeGa (a) and Co2FeSi (b) at the Fermi level region.

Interesting properties of these materials result from exchange interactions between electrons and holes

near bands. In a tetrahedral crystal field, the d states divide into two states, t2g and eg , while the p states have

a symmetry t2g . States with the same symmetry form strong p-d hybridization. This hybridization reduces the

magnetic moment.

According to Galanakis et al. [59], the total magnetic moment in Heusler alloys follows the rule of Slater

and Pauling, mt = Nv – 24, where Nv is the total number of valence band electrons, even for compounds

containing more than 24 electrons such as the alloys studied in our case, Co2FeGa (Nv = 29) and Co2FeSi

(Nv = 30).

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As we can see in Figure 3, which represents the total magnetic moment in function of the number

of valance band electrons (Slater–Pauling behavior), our calculations show a small deviation from this rule,

indicating that some compounds are not perfectly semimetals. Heusler cobalt-based alloys are also intermetallic

compounds based on 3d transition metals and they present a localized magnetism relative to a route character.

The explanation of the magnetism origin of these alloys is very complicated; however, their magnetic moments

vary regularly according to the number of valence band electrons and the crystalline structure. This behavior

is called Slater–Pauling behavior [60,61]. Those two physicists discovered that the magnetic moment m of 3d

elements can be estimated on the basis of the average number of valence electrons (Nv) per atom.

6 7 8 9 10 11

0.0

0.5

1.0

1.5

2.0

2.5

Co

Ni

Fe

Cr

fcc

itinerant

bcc

localised

Co2 TiAl

Co2 VAl

Co2 CrGa

Co2 MnAl

Co2 MnSi

Co2 FeSi

Heusler

FeCo

FeCr

FeV

Mag

net

ic m

om

ent

per

ato

m m

[µ

B]

Valence electrons per atom nV

FeCo

FeNi

NiCo

NiCu

CoCr

NiCr

Figure 3. The Slater–Pauling curve for the 3d alloys in function of the number of valence band electrons.

The materials in Figure 3 are divided into two areas according to m (Nv): the first zone of the Slater–

Pauling curve is the domain of low valence electron concentrations (Nv ≤ 8) and localized magnetism. Here,

the related structures mainly found are bcc. The second zone is the zone of high valence electron concentrations

(Nv ≥ 8) and itinerant magnetism. In this area, systems with closed structures are found such as fcc. Iron (Fe)

is located on the border between localized and itinerant magnetism. However, Heusler alloys are located in the

localized part of this curve. Therefore, we focus on this part of the curve. The magnetic moment per atom is

given by the following relation:

m ≈ Nv − 6, (3)

where Nv is the number of valence band electrons.

As presented before, cobalt-based Heusler alloys show eight d minority states contained in the lattice.

There is a doubly degenerate state of lower energy eg , a triply degenerate state of lower energy t2g , and a triply

degenerate state of higher energy t1u below the Fermi level. Besides the state d, there is a state s and three

other p states, which are not counted in the gap structure. Finally, we have twelve minority occupied states per

lattice. The total magnetic moment is given by the majority electron number in excess (Nmax) with relative to

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minority electrons (Nmin):

m = Nmaj −Nmin. (4)

The number of valence band electrons is determined by:

Nv = Nmaj +Nmin, (5)

and the total magnetic moment becomes:

m = Nv − 24. (6)

This relation is called the generalized Slater–Pauling rule, equivalent to the Slater–Pauling behavior for binary

alloys of transition metals [62]. Since cobalt-base Heusler alloys have an integer number of valence, this rule

is used to determine their magnetic moments. Figure 3 shows that the Heusler alloys magnetic moment is

controlled by the atom Z. For example, the Si, which has four valence electrons, has a higher magnetic moment

than an equivalent Heusler compound that contains Ga as the Z element. This effect is due to the increase of

the d electron number associated with the Z atom. As indicated before, the structural changes of the Heusler

alloys can have a significant effect on their magnetic properties. Also, all atomic exchanges can change the

local hybridization of orbitals. The magnetic moment from valence electrons located at the orbital d level can

be affected by this interatomic exchange. The change of the lattice parameter can be used to describe the

structural disorder level. For instance, the alloy Co2FeSi presents a magnetic moment of 6 µB /f.u. for the

phase L21 [63]. In the case of the exchange between the atoms Co and Fe, the magnetic moment is reduced to

5.5 µB /f.u., whereas, during the exchange between Co and Si, the disorder leads to an increase in the magnetic

moment at 6.05 µB /f.u.

These characteristics of magnetic moment in the case of atom exchange and the behavior of spins in these

two Heusler alloys lead to the appearance of new quantum states. Each spin state represents a quantum bit. In

contrast to the classical bit, which is a fundamental concept of classical computation that occupies two states

(0 and 1), the quantum bit (Qubit) exists in many different states according to the spin polarization and state

shown before. The Qubit is a fundamental concept of quantum computation, which can be in a superposition

of states |0⟩ and |1⟩ , where |0⟩ and |1⟩ are known as computational basis states. In our case they represent

the two spin up and down states of polarization. This is the Dirac representation of classical states 0 and 1,

respectively. Thus, we can represent a Qubit state as a linear combination of states |0⟩ and |0⟩ as follows:

|ψ⟩ = α|0⟩+ β|1⟩ (7)

This shows that a state |ψ⟩ is in superposition of states |0⟩ and |1⟩ , where α andβ are complex numbers

representing the probability of states spin up and spin down: |α|2 + |β|2 = 1. Therefore, we cannot precisely

determine the final Qubit state. However, quantum mechanics can measure a Qubit state. This gives either |0⟩

with probability |α|2or |1⟩ with probability |β|2 . Controlling the spin behavior of the two Heusler compounds

Co2FeGa and Co2FeSi leads to total control over Qubits. As a result, we can construct quantum gates. Similar

to the way a classical computer is built from an electrical circuit containing logic gates (AND, OR, NOT. . . etc.)

and wires, a quantum computer is built from a quantum circuit containing quantum gates (QNOT, Hadamard,

Pauli-Y, Pauli-Z, etc.) and wires to perform quantum computation and information.

6. Conclusion

In this work we focus on the study of a very precise class of ferromagnetic Heusler alloys of the type Co2FeZ,

where Z is either Ga or Si. Our calculations are based on DFT with the help of WIEN2K code, which is an

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implementation of the FP-LAPW method. The exchange and the correlation potential is treated with GGA as

presented by Perdew et al. [50–53] . Due to their magnetic and electronic properties, these materials seem to

be much more practical than other classes of ferromagnetic alloys for quantum computation applications.

The results showed that the total magnetic moment for these alloys is between 5.5 µB /f.u. and 6.5

µB /f.u., which is in perfect agreement with Slater and Pauling’s rulemt = Nv – 24. The spin polarization at

the Fermi level is high, and it can reach up to 100%. Moreover, the structural properties such as DOS clearly

show the metallic character of these alloys given by the absence of a gap at the Fermi level.

The structural changes in Heusler alloys can have a significant effect on their magnetic properties. All

these atomic exchanges can change the local hybridization of orbitals. The magnetic moments from the valence

electrons that localize in the level of orbitals d can be affected by this interatomic exchange. The lattice

parameter change can serve to describe the structural disorder. For example, the Co2FeSi alloy presents a

magnetic moment of 6 µB /f.u. for the L21 phase. In the case of the exchange between the Co and Fe atoms,

the magnetic moment is reduced to 5.5 µB /f.u. However, in the case of the exchange between Co and Si atoms,

the disorder increases the magnetic moment to 6.05 µB/ f.u. The calculations prove that Co2FeSi and Co2FeGa

are semimetallic.

With regard to the two compounds, we remark that the metallic characteristic is dominant in both

directions of the spin, i.e. there is no band gap at the Fermi level whether it is in the direction of majority spins

or that of minority spins. This confirms that it is not quite half-metallic. However, this compound does not

have perfect half metallic characteristics; a small change of the lattice parameter can restore the half metallicity

and can give a full magnetic moment. Indeed, the variation of the lattice constant modifies the hybridization

between the electrons of the different atoms. This variation causes the change of the gap width and position

and the electrons become more localized, which results in a shift of the majority and the minority bands relative

to the Fermi level. The bandwidth of the minority bands in both compounds consists essentially of covalent

hybridization between the d states of Co and Fe, leading to the binding and antibinding band formations with

a gap between the two.

Due to the lattice parameter variation, the two Heusler compounds (Co2FeGa and Co2FeSi) have

approximately a complete (100%) spin polarization at the Fermi level. These properties widely serve the

spintronics and quantum computation fields. Furthermore, they attracted much attention for their useful

applications in spin-dependent devices. In quantum computation we start to invest in these structural and

electronic properties to make a new generation of quantum devices such as spin injection devices, nonvolatile

magnetic random access memories, and magnetic Qubits. Doing that, we depend on the two spin states (spin

up and spin down) to determine the basic element for quantum computation, which is the Qubit. Full control

over the spin states will contribute to the construction of a set of universal quantum gates and circuits totally

different than the classical one based on semiconductor technology. As a final result, the fabrication of a future

quantum computer will be possible.

7. Acknowledgments

We are grateful to all of those with whom we have had the pleasure to work during this and other related

papers. Also, the authors appreciate the referee’s valuable and profound comments that greatly improved the

manuscript.

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References

[1] Semiconductor Industry Association. National Technology Roadmap for Semiconductors 1997. San Jose, CA, USA:

SIA, 1997.

[2] Moore G. Cramming more circuits onto integrated circuits. Electronics 1965; 38: 114-117.

[3] Burger D, Goodman J. Billion-transistor architectures: there and back again. Computer 2004; 37: 22-28.

[4] Meindl JD, Davis JA. The fundamental limit on binary switching energy for terascale integration (TSI). IEEE J

Solid-St Circ 2000; 35: 1515-1516.

[5] Hennessy JL, Patterson D. Computer Architecture: A Quantitative Approach. 3rd ed. Amsterdam, the Netherlands:

Morgan Kaufman, 2003.

[6] Ling J. In: Future Horizons Industry Midterm Forecast Seminar; 20 July 2010; Kensington, London. p. 36.

[7] Murari B. Integrating nanoelectronic components into electronic microsystems. IEEE Micro 2003; 23: 36-44.

[8] Ronen R, Mendelson A, Lai K, Pollack F, Shen J. Coming challenges in microarchitecture and architecture. P IEEE

2001; 83: 325-339.

[9] Claassen T. System on chip: changing IC design today and in the future. IEEE Micro 2004; 23: 20-26.

[10] Plummer J, Griffin P. Material and process limits in silicon VLSI technology. P IEEE 2001; 89: 240-258.

[11] Benioff P. Quantum mechanical Hamiltonian models of Turing machines. J Stat Phys 1982; 29: 515-546.

[12] Feynman R. Quantum mechanical computers. Optics News 1985; 11: 11-20.

[13] Feynman R. Simulating physics with computers. Int J Theor Phys 1982; 21: 467-488.

[14] Deutsch D, Jozsa R. Rapid solution of problems by quantum computation. P R Soc London 1992; A439: 553-558.

[15] Simon D. On the power of quantum computation. SIAM J Comput 1997; 26: 1474-1483.

[16] Shor P. Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual

IEEE Symposium on Foundations of Computer Science; 1994. New York, NY, USA: IEEE. pp. 124-134.

[17] Shor P. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM

J Comput 1997; 26: 1484-1509.

[18] Vandersypen L, Steffen M, Breyta G, Yannoni C, Cleve R, Chuang I. Experimental realization of an order-finding

algorithm with an NMR quantum computer. Phys Rev Lett 2000; 85: 5452-5455.

[19] Deutsch D. Quantum theory, the Church–Turing principle and the universal quantum computer. P R Soc London

1985; A400: 97-117.

[20] Knill E. Approximation by Quantum Circuits. Los Alamos National Laboratory Technical Report. Los Alamos,

NM, USA: Los Alamos National Laboratory, 1995.

[21] Kitaev A, Shen A, Vyalyi M. Classical and Quantum Computation. Boston, MA, USA: American Mathematical

Society, 2002.

[22] Nielsen M, Chuang I. Quantum Computation and Quantum Information. 2nd ed. New York, NY, USA: Cambridge

University Press, 2010.

[23] Abutaleb M. A new static differential design style for hybrid SET CMOS logic circuits. J Comput Electron 2015;

14: 329-340.

[24] Fenner S, Green F, Homer S, Zhang Y. Bounds on the power of constant-depth quantum circuits. In: Proceedings

of the 15th International Symposium on Fundamentals of Computation Theory; 2005. pp. 44-55.

[25] Bernstein E, Vazirani U. Quantum complexity theory. SIAM J Comput 1997; 26: 1411-1473.

[26] Urias J, Quinones D. House holder methods for quantum circuit design. Can J Phys 2015; 94: 150-158.

1258


[27] Jia P, Tian F, Fan S, He Q, Feng J, Yang S. A novel sensor array and classifier optimization method of electronic

nose based on enhanced quantum-behaved particle swarm optimization. Sensor Rev 2014; 34: 304-311.

[28] Cleve R, Palma G, Zeilinger A. An introduction to quantum complexity theory. In: Macchiavello C, Palma GM,

Zeilinger A. Collected Papers on Quantum Computation and Quantum Information Theory. Singapore: World

Scientific, 2000. pp. 103-127.

[29] Rosgen B, Watrous J. The hardness of distinguishing mixed-state quantum computations. In: Proceedings of the

20th Annual Conference on Computational Complexity; 2005. pp. 344-354.

[30] Zurek WH. Decoherence, einselection, and the quantum origins of the classical. Rev Mod Phys 2003; 75: 715

[31] Doplicher S. The measurement process in local quantum theory and the EPR paradox. arXiv: 0908.0480v1, 2012.

[32] Blinder SM. Introduction to Quantum Mechanics: In Chemistry, Materials Science, and Biology. Amsterdam, the

Netherlands: Elsevier, 2004.

[33] Kuhn TS. The Structure of Scientific Revolutions. 3rd ed. Chicago, IL, USA: University of Chicago Press, 1996.

[34] Paz JP, Zurek WH. Environment-induced decoherence and the transition from quantum to classical. Lect Notes

Phys 2002; 587: 77-148.

[35] Jacak L, Hawrylak P, Wojs A. Quantum Dots. Berlin, Germany: Springer, 1998.

[36] Awschalom D, Loss D, Samarth N. Semiconductor Spintronics and Quantum Computation. Berlin, Germany:

Springer, 2002.

[37] Slipko VA, Savran I, Pershin YV. Spontaneous emergence of a persistent spin helix from homogeneous spin

polarization. Phys Rev B 2011; 83: 193302.

[38] Kodera T, Vanderwiel WG, Maruyama T, Hirayama Y, Tarucha S. Fabrication and characterization of quantum

dot single electron spin resonance devices. In: Proceedings of the International Symposium on Mesoscopic Super-

conductivity and Spintronics — In the Light of Quantum Computation; 2004. pp. 445-450.

[39] Muthukrishnan A, Stroud CR. Multi-valued logic gates for quantum computation. arXiv: quant-ph/0002033v2,

2000.

[40] Dreizler RM, Gross E, Parrandw R. Density Functional Theory: An Approach to the QuantumMany-Body Problem.

Berlin, Germany: Springer-Verlag, 1990.

[41] Gabor MS, Tiusan C, Petrisor T, Petrisor T, Hehn M, Lu Y, Snoeck E. Structural defects analysis versus spin

polarized tunneling in Co2FeAl/MgO/CoFe magnetic tunnel junctions with thick MgO barriers. J Magn Magn

Mater 2013; 347: 79-85.

[42] Trudel S, Gaier O, Hamrle J, Hillebrands B. Magnetic anisotropy, exchange and damping in cobalt-based full-Heusler

compounds: an experimental review. J Phys D Appl Phys 2010; 43: 193001.

[43] Kallmayer M, Conca A, Jourdan M, Schneider H, Jakob G, Balke B, Gloskovski A, Elmers HJ. Correlation of local

disorder and electronic properties in the Heusler alloy Co2Cr0.6Fe0.4Al. J Phys D Appl Phys 2007; 40: 1539-1543.

[44] Belmeguenai M, Tuzcuoglu H, Cherif SM, Westerholt K, Chauveau T, Mazaleyrat F, Moch P. Cu2MnAl thin films

grown onto sapphire and MgO substrates: exchange stiffness and magnetic anisotropy. Phys Status Solidi A 2013;

210: 553-558.

[45] Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys Rev B 1964; 136: B864.

[46] Blaha P, Schwarz K, Sorantin P, Tricky S. First principle investigations on electronic, magnetic, thermodynamic,

and transport properties of TlGdX2 (X = S, Se, Te). Comput Phys Commun 1990; 59: 399.

[47] Blaha P, Schwarz K, Madsen GK, Kvasnicka D, Luitz J, Schwarz K. WIEN2K: An augmented plane wave plus local

orbitals program for calculating crystal properties. Vienna, Austria: Vienna Technical University, 2001.

[48] Loucks TL. The Augmented Plane Wave Method: A Guide to Performing Electronic Structure Calculations. New

York, NY, USA: W.A. Benjamin, 1967.

[49] Takeda T. Linear methods for fully relativistic energy-band calculations. J Phys F Met Phys 1979; 9: 815.

1259


[50] Perdew JP, Wang Y. Accurate and simple density functional for the electronic exchange energy: generalized gradient

approximation. Phys Rev B 1986; 33: 8800.

[51] Perdew JP, Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev

B 1992; 45: 13244.

[52] Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys Rev Lett 1996; 77: 3865.

[53] Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys Rev Lett 1997; 78: 1396.

[54] Monkhorst HJ, Pack JD. Special points for Brillonin-zone integrations. Phys Rev B 1976; 13: 5188.

[55] Hehn M. Magnetisme et transport polarise en spin : de la couche mince aux dispositifs. Matiere Condensee. Nancy,

France: Universite Henri Poincare, 2004.

[56] Wurmehl S, Fecher GH, Kandpal HC, Ksenofontov V, Felser C, Lin HJ, Morais J. Properties of the quaternary

half-metal-type Heusler alloy Co2Mn1−xFexSi. Phys Rev B 2006; 74: 104405.

[57] Wurmehl S, Fecher GH, Kandpal HC, Ksenofontov V, Felser C, Lin HJ. Investigation of Co2FeSi: the Heusler

compound with highest Curie temperature and magnetic moment. Appl Phys Lett 2006; 88: 032503.

[58] Fecher GH, Felser C, Kubler J. Understanding the trend in the Curie temperatures of Co2 -based Heusler compounds:

Ab initio calculations. Phys Rev B 2007; 76: 024414.

[59] Galanakis I, Dederichs PH, Papanikolaou N. Origin and properties of the gap in the half-ferromagnetic Heusler

alloys. Phys Rev B 2002; 66: 134428.

[60] Slater J. The ferromagnetism of nickel. Phys Rev 1936; 49: 537-545.

[61] Pauling L. The nature of the interatomic forces in metals. Phys Rev 1938; 54: 899-904.

[62] Kubler J, William A, Sommers C. Formation and coupling of magnetic moments in Heusler alloys. Phys Rev B

1983; 28: 1745-1755.

[63] Kandpal H, Fecher G, Felser C, Schonhense G. Correlation in the transition-metal-based Heusler compounds

Co2MnSi and Co2FeSi. Phys Rev B 2006; 73: 094422.

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